"Mail Supremacy" by Hayford Pierce. As per this review:
Hayford Peirce’s “Mail Supremacy” is a short, light-hearted story in which Chap Foey Rider begins to wonder about the mail system and how it works. Rider, who runs an import company in New York, laments the loss of multiple deliveries a day and further notes that it seems that something mailed from a shorter distance takes longer to reach its destination than something mailed from a longer distance. He is more likely to receive a letter from his office in Los Angeles first than a letter mailed from nearby Boston.
He begins to test this by having his office managers mail letters and tracking their time in transit. Once he is sure that letters mailed far distances are being delivered quickly, he takes it to the illogical extreme and tries to mail letters to Alpha Centauri. “Mail Supremacy” doesn’t take itself seriously at all and in some ways is a satire on the idea of a Galactic Federation, even as it served Peirce as a starting point for his own series of stories about a Galactic Federation.